Simple noise-reduction method based on nonlinear forecasting

Nonparametric detrending or noise reduction methods are often employed to separate trends from noisy time series when no satisfactory models exist to fit the data. However, conventional noise reduction methods depend on subjective choices of smoothing parameters. Here we present a simple multivariate noise reduction method based on available nonlinear forecasting techniques. These are in turn based on state-space reconstruction for which a strong theoretical justification exists for their use in nonparametric forecasting. The noise reduction method presented here is conceptually similar to Schreiber's noise reduction method using state-space reconstruction. However, we show that Schreiber's method has a minor flaw that can be overcome with forecasting. Furthermore, our method contains a simple but nontrivial extension to multivariate time series. We apply the method to multivariate time series generated from the Van der Pol oscillator, the Lorenz equations, the Hindmarsh-Rose model of neuronal spiking activity, and to two other univariate real-world data sets. It is demonstrated that noise reduction heuristics can be objectively optimized with in-sample forecasting errors that correlate well with actual noise reduction errors.

Figure 1

Performance of the noise reduction method. Error bars are estimates of standard errors. (a) The phase portrait of the noisy Van der Pol oscillator. (b) The phase portrait of the noise-reduced Van der Pol oscillator in red, including the deterministic Van der Pol oscillator in black. The corrected time series were calculated with the bidirectional algorithm for five and four recursive iterations for the $x$ and $y$ coordinates, respectively. These optimized parameters (for the $x$ coordinate) were determined with (c), the in-sample cross-validation forecasting error for the $x$ coordinate versus the recursive iteration number $r$. An in-sample forecasting error for a time series cleaned $r-1$ times is associated with the potential noise reduction performance of the time series cleaned $r$ times. Therefore, a data point at $r$ is the in-sample forecasting error of a time series that has been cleaned $r-1$ times. (d) The actual noise reduction errors (MAE) of the corrected time series as calculated from ${\mathbf{X}}_t$. Also indicated on the plot is $MAE\left({\epsilon }_t\right)$, the MAE of the noisy time series as calculated from ${\mathbf{X}}_t$. [(e)–(h)] Same as (a) to (d) but for the $x$ coordinate of the Lorenz system. Corrected time series were calculated with the bidirectional algorithm with four recursive iterations. Time series are shown for ($\mathbf{E}$ and $\mathbf{F}$) instead of phase portraits but it should be noted that noise reduction was conducted concurrently for all three variables of the Lorenz system. [(i)–(l)] Same as (e) to (h) but for the $x$ coordinate of the Hindmarsh-Rose model. Noise-reduced time series were calculated with the bidirectional algorithm with three recursive iterations.

Figure 2

Noise-reduced time series of the Lorenz $x$ coordinate using Schreiber's method. Noisy time series from the Lorenz $x$ coordinate [Fig. 1] were noise reduced with Schreiber's method [12]. The resulting noise reduced time series (shown in red) and the actual deterministic time series (shown in black) are plotted in this figure.

Figure 3

Out-of-sample forecast performance of noise-reduced time series. The out-of-sample forecast performance using the original noisy time series and noise-reduced time series as libraries for forecasting. Noise-reduced time series were objectively corrected with the noise reduction method. Forecast performance is measured by the normalized MAE for the various systems. Forecast MAEs (blue and red bars) are normalized with the MAE of the dynamical trend (black bars) which is calculated against the noisy out-of-sample time series. For the measles data set and global surface temperature data set (Temp), MAEs are normalized against the MAE from the noise-reduced time series instead due to an unknown dynamical trend. Error bars are estimates of standard errors.